Identify distribution from PGF

Given a PGF in standard form, identify the underlying distribution and state its parameters.

1 questions

Edexcel FS1 2023 June Q6
  1. The discrete random variable \(X\) has probability generating function
$$\mathrm { G } _ { X } ( t ) = \frac { t ^ { 2 } } { ( 3 - 2 t ) ^ { 2 } }$$
  1. Specify the distribution of \(X\) A fair die is rolled repeatedly.
  2. Describe an outcome that could be modelled by the random variable \(X\)
  3. Use calculus and \(\mathrm { G } _ { X } ( t )\) to find
    1. \(\mathrm { E } ( X )\)
    2. \(\operatorname { Var } ( X )\) The discrete random variable \(Y\) has probability generating function $$\mathrm { G } _ { Y } ( t ) = \frac { t ^ { 10 } } { \left( 3 - 2 t ^ { 3 } \right) ^ { 2 } }$$
  4. Find the exact value of \(\mathrm { P } ( Y = 19 )\)